Question

To perform an analysis of a mixture of metal ions by electro-deposition, the second metal to be deposited must not being plating out until the concentration ratio of the second to the first is about \(10^{6}\). What must be the minimum difference in standard potential of the two metals which form dipositive ions in order for such an analysis to be feasible? (a) \(0.177 \mathrm{~V}\) (b) \(0.354 \mathrm{~V}\) (c) \(0.708 \mathrm{~V}\) (d) \(0.088 \mathrm{~V}\)

Short answer

The minimum difference in standard potential of the two metals which form dipositive ions, in order for such an analysis to be feasible, has to be at least 0.177 V.

Step-by-step solution

Understand the Nernst Equation

The Nernst equation relates the reduction potential of an electrochemical cell to the standard reduction potential, the temperature, the number of moles of electrons transferred in the redox reaction, and the reaction quotient. It is given by: \[\begin{equation} E = E^0 - \frac{0.0592}{n} \log Q \end{equation}\] where, - \(E\) is the cell potential, - \(E^0\) is the standard cell potential, - \(n\) is the number of moles of electrons transferred, - \(Q\) is the reaction quotient i.e., activity or concentration of products over reactants.For the reaction to begin at an appreciable rate, the potential must overcome the overpotential as well.

Apply the Concentration Ratio to the Nernst Equation

Let \(E_1^0\) and \(E_2^0\) be the standard potentials of the first and second metal, respectively. The cell potential for the second metal \(E_2\) to start depositing can be expressed using the Nernst equation:\[\begin{equation} E_2 = E_2^0 - \frac{0.0592}{2} \log \frac{[M^{2+}]_2}{1} \end{equation}\]Since we want the concentration of the second metal to be about \(10^6\) times that of the first before it starts to deposit, we can express the potential for the first metal \(E_1\) when this condition is met:\[\begin{equation} E_1 = E_1^0 - \frac{0.0592}{2} \log \frac{[M^{2+}]_1}{1} \end{equation}\]where \([M^{2+}]_2 = 10^6 [M^{2+}]_1\).

Calculate the Minimum Difference in Standard Potentials

To prevent the second metal from plating out, its cell potential must be less than or equal to the cell potential for the first metal. Thus:\[\begin{equation} E_2 \leq E_1 \end{equation}\]Substituting the Nernst equation: \[\begin{equation} E_2^0 - \frac{0.0592}{2} \log \frac{[M^{2+}]_2}{1} \leq E_1^0 - \frac{0.0592}{2} \log \frac{[M^{2+}]_1}{1} \end{equation}\]Since \([M^{2+}]_2 = 10^6 [M^{2+}]_1\), the inequality becomes:\[\begin{equation} E_2^0 - \frac{0.0592}{2} \log 10^6 \leq E_1^0 \end{equation}\]Solving this gives: \[\begin{equation} E_2^0 - E_1^0 \leq \frac{0.0592}{2} \times 6 \end{equation}\]Which simplifies to: \[\begin{equation} \Delta E^0 \leq 0.177V \end{equation}\]Therefore, the minimum difference in standard potential that would prevent the second metal from plating out until the concentration ratio reaches about \(10^6\) is 0.177V.

Key concepts

Nernst Equation

The Nernst equation is a fundamental formula in electrochemistry that predicts the electromotive force of a reaction based on the inherent chemistry and the conditions under which it is taking place. It integrates the thermodynamics of reduction and oxidation reactions with practical observations. Simply put, it helps chemists and engineers understand how the voltage of an electrochemical cell will change as the concentration of reactants and products shifts over time.

For an electrochemical cell undergoing a reversible reaction, the Nernst equation in its general form is expressed as:
\[\begin{equation}E = E^0 - \frac{0.0592}{n} \log Q\end{equation}\]
where:

• \(E\) is the potential of the reaction under non-standard conditions,
• \(E^0\) is the standard reduction potential,
• \(n\) represents the number of electrons transferred in the half-reaction,
• and \(Q\) is the reaction quotient, a measure of the ratio of concentrations of products to reactants at any point during the reaction process.

Standard Reduction Potential

The standard reduction potential, denoted as \(E^0\), is an intrinsic property of a substance that measures its tendency to gain electrons and thus be reduced. This value is determined under standard conditions, which includes a temperature of 298K, 1M concentration for each ion participating in the reaction, a pressure of 1 bar for gases, and pure solids or liquids in their standard states.

Substances with a higher standard reduction potential have a greater tendency to undergo reduction, that is, to accept electrons. When we compare two different half-reactions, the one with the higher standard reduction potential will act as the cathode and attract electrons, while the one with the lower standard reduction potential will act as the anode and lose electrons.

The importance of these potentials lies in their predictive power; they can forecast the spontaneous direction of an electrochemical reaction. This is crucial for performing electrodeposition analysis, as knowing the potentials allows us to control which metal ions will deposit first.

Electrochemical Cell Potential

The electrochemical cell potential, also known as the electromotive force (emf) of the cell, is the total voltage that an electrochemical cell can produce under specific conditions. It is determined by the difference in standard reduction potential between the two half-cells that make up the electrochemical system. Mathematically, the cell potential is given by:\[E = E_{cathode}^0 - E_{anode}^0\]
Understanding the cell potential is essential for practical applications, such as calculating the energy that can be harnessed from a battery or the voltage required to drive an electrolysis reaction. In the context of electroplating, the cell potential gives insights into how metals will be deposited, dictating the order in which different metal ions will plate out onto the cathode based on their individual reduction potentials.

Concentration Ratio

The concentration ratio, in the context of electrodeposition, refers to the ratio of the concentrations of metal ions in the solution. It plays a pivotal role in determining the order and rate at which different metal ions are deposited onto the cathode during the plating process.

A greater concentration ratio means that there is a higher preference for one metal ion to be deposited over another. This is why in an analysis of a mixture of metal ions, the concentration ratio can provide control over the separation and quantification of those ions based on their deposition.In the exercise given, a concentration ratio of approximately \(10^6\) is desired to ensure selective deposition, which means that the desired metal should be present at a concentration a million times higher than the competing metal ion before the latter starts to deposit. Such criteria ensure that highly precise electrochemical analysis and purification processes can be achieved.